Skip to main content

Benchmarking Solvers for the One Dimensional Cubic Nonlinear Klein Gordon Equation on a Single Core

  • Conference paper
  • First Online:
Benchmarking, Measuring, and Optimizing (Bench 2019)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12093))

Included in the following conference series:

  • 1073 Accesses

Abstract

To determine the best method for solving a numerical problem modeled by a partial differential equation, one should consider the discretization of the problem, the computational hardware used and the implementation of the software solution. In solving a scientific computing problem, the level of accuracy can also be important, with some numerical methods being efficient for low accuracy simulations, but others more efficient for high accuracy simulations. Very few high performance benchmarking efforts allow the computational scientist to easily measure such tradeoffs in order to obtain an accurate enough numerical solution at a low computational cost. These tradeoffs are examined in the numerical solution of the one dimensional Klein Gordon equation on single cores of an ARM CPU, an AMD x86-64 CPU, two Intel x86-64 CPUs and a NEC SX-ACE vector processor. The work focuses on comparing the speed and accuracy of several high order finite difference spatial discretizations using a conjugate gradient linear solver and a fast Fourier transform based spatial discretization. In addition implementations using second and fourth order timestepping are also included in the comparison. The work uses accuracy-efficiency frontiers to compare the effectiveness of five hardware platforms

BKM was partially supported by HPC Europa 3 (INFRAIA-2016-1-730897). Compute time on Isamabard was partially supported by ESPRC grant EP/P020224/1.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdulkadir, Y.A.: Comparison of finite difference schemes for the wave equation based on dispersion. J. Appl. Math. Phys. 3, 1544–1562 (2015). https://doi.org/10.4236/jamp.2015.311179

    Article  Google Scholar 

  2. Adams, M.F., Brown, J., Shalf, J., Van Straalen, B., Strohmaier, E., Williams, S.: HPGMG 1.0: A Benchmark for Ranking High Performance Computing Systems, Lawrence Berkely National Laboratory Preprint (2014). https://escholarship.org/uc/item/00r9w79m. Accessed 16 July 2019

  3. Afanasyev, I.V., et al.: Developing efficient implementations of Bellman-Ford and Forward-Backward Graph Algorithms for NEC SX-ACE. Supercomput. Front. Innov. 5(3), 65–69 (2018). https://doi.org/10.14529/jsfi180311

    Article  Google Scholar 

  4. Arm Performance Library. https://www.arm.com/products/development-tools/server-and-hpc/allinea-studio/performance-libraries. Accessed 16 Nov 2019

  5. Aseeri, S., et al.: Solving the Klein-Gordon equation using Fourier spectral methods: a benchmark test for computer performance. In: HPC 2015 Proceedings of the Symposium on High Performance Computing, pp. 182–191. Society for Computer Simulation International (2015)

    Google Scholar 

  6. Aseeri, S., Muite, B.K., Takahashi, D.: Reproducibility in benchmarking parallel fast Fourier transform based applications. In: Companion of the 2019 ACM/SPEC International Conference on Performance Engineering - ICPE 2019, pp. 5–8 (2019). https://doi.org/10.1145/3302541.3313105

  7. Auzinger, W., Br̆ezinová, I., Hofstätter, H., Koch, O., Quell, M.: Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part II: comparisons of local error estimation and step-selection strategies for nonlinear Schrödinger and Wave equations. Comput. Phys. Commun. 234, 55–71 (2018). https://doi.org/10.1016/j.cpc.2018.08.003

  8. Bailey, D.H., et al.: The NAS parallel benchmarks. Int. J. High Perform. Comput. Appl. 5(3), 63–73 (1991). https://doi.org/10.1177/109434209100500306

    Article  Google Scholar 

  9. Balakrishnan, S., et al.: Parallel Spectral Numerical Methods. http://en.wikibooks.org/wiki/Parallel_Spectral_Numerical_Methods. Accessed 24 June 2019

  10. Buttari, A., Dongarra, J., Kurzak, J., Luszczek, P., Tomov, S.: Using mixed precision for sparse matrix computations to enhance performance while achieving 64-bit accuracy. ACM Trans. Math. Softw. 34(4), 17 (2008). https://doi.org/10.1145/1377596.1377597

    Article  MathSciNet  MATH  Google Scholar 

  11. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-540-30726-6

    Book  MATH  Google Scholar 

  12. Cloutier, B., Muite, B.K., Parsani, M.: Fully implicit time stepping can be efficient on parallel computers. Supercomput. Front. Innov. 6(2), 75–85 (2019). https://doi.org/10.14529/jsfi190206

    Article  Google Scholar 

  13. Cloutier, B., Muite, B.K., Rigge, P.: A comparison of CPU and GPU performance for Fourier Pseudospectral Simulations of the Navier-Stokes, Cubic Nonlinear Schrödinger and Sine Gordon Equations. In: Proceedings of the 2012 Symposium on Application Accelerators in High Performance Computing, pp. 145–148 (2012). https://doi.org/10.1109/SAAHPC.2012.24

  14. Chang, J., Nakshatrala, K.B., Knepley, M.G., Johnsson, L.: A performance spectrum for parallel computational frameworks that solve PDEs. Concurr. Comput. Pract. Exp. 30, e4401 (2018). https://doi.org/10.1002/cpe.4401

    Article  Google Scholar 

  15. Deconinck, W., et al.: Accelerating extreme-scale numerical weather prediction. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds.) PPAM 2015. LNCS, vol. 9574, pp. 583–593. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-32152-3_54

    Chapter  Google Scholar 

  16. Dongarra, J., Heroux, M.A., Luszcek, P.: A new metric for ranking high-performance computing systems. Int. J. High Perform. Comput. Appl. 30(1), 3–10 (2016). https://doi.org/10.1177/1094342015593158

    Article  Google Scholar 

  17. Frigo, M., Johnson, S.G.: The design and implementation of FFTW. Proc. IEEE 93(2), 216–231 (2005). https://doi.org/10.1109/JPROC.2004.840301

    Article  Google Scholar 

  18. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press (1996). https://doi.org/10.1017/CBO9780511626357

  19. Fornberg, B.: Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput. 51, 699–706 (1988). https://doi.org/10.1090/S0025-5718-1988-0935077-0

    Article  MathSciNet  MATH  Google Scholar 

  20. Gholami, A., Malhotra, D., Sundar, H., Biros, G.: FFT, FMM, or Multigrid? A comparative study of state-of-the-art poisson solvers for uniform and nonuniform grids in the unit cube. SIAM J. Sci. Comput. 38(3), C280–C306 (2016). https://doi.org/10.1137/15M1010798

    Article  MathSciNet  MATH  Google Scholar 

  21. GW4: Isambard. https://gw4.ac.uk/isambard/. Accessed 9 Nov 2019

  22. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations I. Springer, Heidelberg (1993). https://doi.org/10.1007/978-3-540-78862-1

    Book  MATH  Google Scholar 

  23. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Heidelberg (1996). https://doi.org/10.1007/978-3-642-05221-7

    Book  MATH  Google Scholar 

  24. Höchstleistungsrechenzentrum Stuttgart (HLRS): Hazelhen. https://www.hlrs.de/systems/cray-xc40-hazel-hen/. Accessed 15 July 2019

  25. Höchstleistungsrechenzentrum Stuttgart (HLRS): Kabuki. https://kb.hlrs.de/platforms/index.php/NEC_SX-ACE. Accessed 15 July 2019

  26. Hutchinson, M., Heinecke, A., Pabst, H., Henry, G., Parsani, M., Keyes, D.: Efficiency of high order spectral element methods on petascale architectures. In: Kunkel, J.M., Balaji, P., Dongarra, J. (eds.) ISC High Performance 2016. LNCS, vol. 9697, pp. 449–466. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41321-1_23

    Chapter  Google Scholar 

  27. Ibeid, H., Olson, L., Gropp, W.: FFT, FMM, and Multigrid on the Road to Exascale: Performance Challenges and Opportunities, arXiv:1810.11883v1 (2018)

  28. Kassam, A.-K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26(4), 1214–1233 (2005). https://doi.org/10.1137/S1064827502410633

    Article  MathSciNet  MATH  Google Scholar 

  29. Ketcheson, D.I., Mortensen, M., Parsani, M., Schilling, N.: More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence. arXiv:1810.10197v1

  30. King Abdullah University of Science and Technology Supercomputing Laboratory: Ibex. https://www.hpc.kaust.edu.sa/ibex. Accessed 9 Nov 2019

  31. Komatitsch, D., et al.: SPECFEM3D Cartesian [software], GITHASH8 (1999). https://geodynamics.org/cig/software/specfem3d/. Accessed 16 July 2019

  32. Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press (2009). https://doi.org/10.1017/CBO9780511614118

  33. McIntosh-Smith, S., Price, J., Poenaru, A., Deakin, T.: Scaling results from the first generation of ARM-based supercomputers. In: Proceedings of the Cray User Group 2019. http://uob-hpc.github.io/assets/cug-2019.pdf. Accessed 9 Nov 2019

  34. Muite, B.K.: https://github.com/bkmgit/KleinGordon1D [software]. Accessed 16 July 2019

  35. Müller, E.H., Scheichl, R., Vainikko, E.: Petascale solvers for anisotropic PDEs in atmospheric modelling on GPU clusters. Parallel Comput. 50, 53–69 (2015). https://doi.org/10.1016/j.parco.2015.10.007

    Article  MathSciNet  Google Scholar 

  36. NEC. http://mathkeisan.com/ [software]. Accessed 16 July 2019

  37. Pershin, I.S., Levchenko, V.D., Perepelkina, A.Y.: Performance limits study of stencil codes on modern GPGPUs. Supercomput. Front. Innov. 6(2), 86–101 (2019). https://doi.org/10.14529/jsfi190207

    Article  Google Scholar 

  38. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-540-71041-7

    Book  MATH  Google Scholar 

  39. Top500. https://www.top500.org/. Accessed 10 Nov 2019

  40. Trefethen, L.: Spectral methods in MATLAB. SIAM 10(1137/1) (2000). https://doi.org/10.1137/1.9780898719598

  41. Treibig, J., Hager, G., Wellein, G.: LIKWID: a lightweight performance-oriented tool suite for x86 multicore environments. In: Proceedings of the First International Workshop on Parallel Software Tools and Tool Infrastructures. https://doi.org/10.1109/ICPPW.2010.38

  42. Williams, S., Waterman, A., Patterson, D.: Roofline: an insightful visual performance model for multicore architectures. Commun. ACM 52(4), 65–76 (2009). https://doi.org/10.1145/1498765.1498785

    Article  Google Scholar 

  43. Yang, C., et al.: 10M-core scalable fully-implicit solver for nonhydrostatic atmospheric dynamics. In: SC 2016: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, pp. 57–68 (2016). https://doi.org/10.1109/SC.2016.5

Download references

Acknowledgements

We thank Holger Berger, José Gracia, John Linford and Simon McIntosh-Smith for helpful conversations. We thank Höchstleistungsrechenzentrum Stuttgart (HLRS), the KAUST Supercomputing Laboratory, the University of Tartu High Performance Computing Center and the GW4 Isamabard project for access to supercomputing resources used in development and testing.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to B. K. Muite .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Muite, B.K., Aseeri, S. (2020). Benchmarking Solvers for the One Dimensional Cubic Nonlinear Klein Gordon Equation on a Single Core. In: Gao, W., Zhan, J., Fox, G., Lu, X., Stanzione, D. (eds) Benchmarking, Measuring, and Optimizing. Bench 2019. Lecture Notes in Computer Science(), vol 12093. Springer, Cham. https://doi.org/10.1007/978-3-030-49556-5_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-49556-5_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-49555-8

  • Online ISBN: 978-3-030-49556-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics